8. Quadrilaterals

Theorem 8.1

A diagonal of a parallelogram divides it into two congruent triangles.

Theorem 8.2

In a parallelogram, opposite sides are equal.

Theorem 8.3

If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.

Theorem 8.4

In a parallelogram, opposite angles are equal.

Theorem 8.5

If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.

Theorem 8.6

The diagonals of a parallelogram bisect each other.

Theorem 8.7

If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Exercise 8.1

1. If the diagonals of a parallelogram are equal, then show that it is a rectangle.
2. Show that the diagonals of a square are equal and bisect each other at right angles.
3. Diagonal AC of a parallelogram ABCD bisects ∠A. Show that it bisects ∠C also, and ABCD is a rhombus.
4. ABCD is a rectangle where diagonal AC bisects both ∠A and ∠C. Show that ABCD is a square and diagonal BD bisects both ∠B and ∠D.

8.2 The Mid-point Theorem

Theorem 8.8

The line segment joining the mid-points of two sides of a triangle is parallel to the third side.

Theorem 8.9

The line drawn through the mid-point of one side of a triangle, parallel to another side, bisects the third side.

Exercise 8.2

1. ABCD is a quadrilateral where P, Q, R, and S are mid-points of its sides. Show that PQRS is a parallelogram.
2. ABCD is a rhombus with mid-points P, Q, R, and S of its sides. Show that PQRS forms a rectangle.
3. ABCD is a rectangle with mid-points P, Q, R, and S of its sides. Show that PQRS is a rhombus.

8.3 Summary